We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrices under the assumption that the absolute values of the off-diagonal matrix entries have uniformly bounded fifth moment and the absolute values of the diagonal entries have uniformly bounded third moment. Using our recent results on the fluctuation of resolvent entries [31], [28] and ideas from [9], we extend the results by Capitaine, and Féral [12], [13].
We study the spectrum of an asymmetric random matrix with block structured
variances. The rows and columns of the random square matrix are divided into
$D$ partitions with arbitrary size (linear in $N$). The parameters of the model
are the variances of elements in each block, summarized in
$g\in\mathbb{R}^{D\times D}_+$. Using the Hermitization approach and by
studying the matrix-valued Stieltjes transform we show that these matrices have
a circularly symmetric spectrum, we give an explicit formula for their spectral
radius and a set of implicit equations for the full density function. We
discuss applications of this model to neural networks
We study the fluctuations of the matrix entries of regular functions of Wigner random matrices in the limit when the matrix size goes to infinity. In the case of the Gaus- sian ensembles (GOE and GUE) this problem was considered by A. Lytova and L. Pastur (J. Stat. Phys. 134:147–159, 2009). Our results are valid provided the off-diagonal matrix entries have finite fourth moment, the diagonal matrix entries have finite second moment, and the test functions have four continuous derivatives in a neighborhood of the support of the Wigner semicircle law. Moreover, if the marginal distributions satisfy the Poincaré inequality our results are valid for Lipschitz continuous test functions
Using a generalized random recurrent neural network model, and by extending our recently developed mean-field approach, we study the relationship between the network connectivity structure and its low dimensional dynamics. Each connection in the network is a random number with mean 0 and variance that depends on pre- and post-synaptic neurons through a sufficiently smooth function g of their identities. We find that these networks undergo a phase transition from a silent to a chaotic state at a critical point we derive as a function of g. Above the critical point, although unit activation levels are chaotic, their autocorrelation functions are restricted to a low-dimensional subspace. This provides a direct link between the network's structure and some of its functional characteristics. We discuss example applications of the general results to neuroscience where we derive the support of the spectrum of connectivity matrices with heterogeneous and possibly correlated degree distributions, and to ecology where we study the stability of the cascade model for food web structure.
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